In the sequential change-point detection literature, most research specifies a required frequency of false alarms at a given pre-change distribution $f_��$ and tries to minimize the detection delay for every possible post-change distribution $g_��$. In this paper, motivated by a number of practical examples, we first consider the reverse question by specifying a required detection delay at a given post-change distribution and trying to minimize the frequency of false alarms for every possible pre-change distribution $f_��$. We present asymptotically optimal procedures for one-parameter exponential families. Next, we develop a general theory for change-point problems when both the pre-change distribution $f_��$ and the post-change distribution $g_��$ involve unknown parameters. We also apply our approach to the special case of detecting shifts in the mean of independent normal observations.

Published at http://dx.doi.org/10.1214/009053605000000859 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)